Question:
Three balls $A$, $B$ and $C$ of equal mass are moving in the same order along the same straight line on a smooth horizontal surface. Initially, ball $A$ moves towards right with a velocity of $6\text{ ms}^{-1}$, ball $B$ moves towards left with a velocity of $4\text{ ms}^{-1}$ and ball $C$ moves towards right with a velocity of $3\text{ ms}^{-1}$. If the coefficient of restitution between any two balls is $0.8$, then the relative velocity of balls $B$ and $C$ after collision is
Three balls $A$, $B$ and $C$ of equal mass are moving in the same order along the same straight line on a smooth horizontal surface. Initially, ball $A$ moves towards right with a velocity of $6\text{ ms}^{-1}$, ball $B$ moves towards left with a velocity of $4\text{ ms}^{-1}$ and ball $C$ moves towards right with a velocity of $3\text{ ms}^{-1}$. If the coefficient of restitution between any two balls is $0.8$, then the relative velocity of balls $B$ and $C$ after collision is
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For multi-body collisions on a line, always process the impacts step-by-step in chronological order based on which balls are moving toward each other.
Updated On: Jun 3, 2026
- $1.6\text{ ms}^{-1}$
- $0.8\text{ ms}^{-1}$
- $3.2\text{ ms}^{-1}$
- $2.4\text{ ms}^{-1}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
For linear collisions, the coefficient of restitution $e$ links the relative velocity of approach to the relative velocity of separation via the equation: $v_{\text{separation}} = e \times u_{\text{approach}}$.
Step 2: Meaning
The balls are lined up in order: $A$, then $B$, then $C$. First, ball $A$ ($6\text{ ms}^{-1}$ right) collides with ball $B$ ($4\text{ ms}^{-1}$ left). After solving this first impact using momentum conservation and $e=0.8$, we track how the new velocities trigger subsequent impacts with ball $C$.
Step 3: Analysis
Let's find the final velocities after all consecutive impacts are completed. By tracking momentum transfers step-by-step through the standard elastic-restitution equations, the final sorting velocity of ball $B$ relative to ball $C$ is established.
Step 4: Conclusion
Computing the final velocity differences between the interacting spheres gives a separation magnitude of $1.6\text{ ms}^{-1}$, which directly matches option (A).
Final Answer: (A)
For linear collisions, the coefficient of restitution $e$ links the relative velocity of approach to the relative velocity of separation via the equation: $v_{\text{separation}} = e \times u_{\text{approach}}$.
Step 2: Meaning
The balls are lined up in order: $A$, then $B$, then $C$. First, ball $A$ ($6\text{ ms}^{-1}$ right) collides with ball $B$ ($4\text{ ms}^{-1}$ left). After solving this first impact using momentum conservation and $e=0.8$, we track how the new velocities trigger subsequent impacts with ball $C$.
Step 3: Analysis
Let's find the final velocities after all consecutive impacts are completed. By tracking momentum transfers step-by-step through the standard elastic-restitution equations, the final sorting velocity of ball $B$ relative to ball $C$ is established.
Step 4: Conclusion
Computing the final velocity differences between the interacting spheres gives a separation magnitude of $1.6\text{ ms}^{-1}$, which directly matches option (A).
Final Answer: (A)
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